Asymmetric inequality for non-homogeneous ternary quadratic forms

Abstract

Let Q( x, y, z) be an indefinite ternary quadratic form of determinant D < 0. Let t≥0 be any given real number. Then the author proves the existence of a function f( t) such that given any reals x 0, y 0, z 0 we can find integers x, y, z such that −t(f(t)|D|) 1 3 < Q(x+x 0,y+y 0,z+z 0) ≤(f(t)|D|) 1 3 . The result is best possible for eight values of t and in particular includes the previous best known results as special cases.